§ 9.9. Zeros

Referenced by:: Ch.9
Permalink:: http://dlmf.nist.gov/9.9

§9.9(i)Distribution and Notation
§9.9(ii)Relation to Modulus and Phase
§9.9(iii)Derivatives With Respect to $k$
§9.9(iv)Asymptotic Expansions
§9.9(v)Tables

§ 9.9(i). Distribution and Notation

Notes:: See Olver (1997b, pp. 414–415).
Permalink:: http://dlmf.nist.gov/9.9.SS1

On the real line, $\mathrm{Ai}\!\left(x\right)$ , ${{\mathrm{Ai}}^{{\prime}}}\!\left(x\right)$ , $\mathrm{Bi}\!\left(x\right)$ , ${{\mathrm{Bi}}^{{\prime}}}\!\left(x\right)$ each have an infinite number of zeros, all of which are negative. They are denoted by $a_{{k}}$ , $a^{{\prime}}_{{k}}$ , $b_{{k}}$ , $b^{{\prime}}_{{k}}$ , respectively, arranged in ascending order of absolute value for $k=1,2,\ldots.$

$\mathrm{Ai}\!\left(z\right)$ and ${{\mathrm{Ai}}^{{\prime}}}\!\left(z\right)$ have no other zeros. However, $\mathrm{Bi}\!\left(z\right)$ and ${{\mathrm{Bi}}^{{\prime}}}\!\left(z\right)$ each have an infinite number of complex zeros. They lie in the sectors $\tfrac{1}{3}\pi<\mathrm{ph}z<\tfrac{1}{2}\pi$ and $-\tfrac{1}{2}\pi<\mathrm{ph}z<-\tfrac{1}{3}\pi$ , and are denoted by $\beta _{{k}}$ , $\beta^{{\prime}}_{{k}}$ , respectively, in the former sector, and by $\bar{\beta _{{k}}}$ , $\bar{\beta^{{\prime}}_{{k}}}$ , in the conjugate sector, again arranged in ascending order of absolute value (modulus) for $k=1,2,\ldots.$ See §9.3(ii) for visualizations.

For the distribution in $\Complex$ of the zeros of ${{\mathrm{Ai}}^{{\prime}}}\!\left(z\right)-\sigma\mathrm{Ai}\!\left(z\right)$ , where $\sigma$ is an arbitrary complex constant, see Muraveĭ (1976).

§ 9.9(ii). Relation to Modulus and Phase

Notes:: See Miller (1946, p. B48) and Olver (1997b, p. 404).
Permalink:: http://dlmf.nist.gov/9.9.SS2

9.9.1 $\theta\!\left(a_{{k}}\right)=\phi\!\left(a^{{\prime}}_{{k+1}}\right)=k\pi,$

Symbols:: $\phi\!\left(z\right)$ : Airy phase function, $\theta\!\left(z\right)$ : Airy phase function, $a_{{k}}$ : $k$ th zero of Airy $\mathrm{Ai}$ , $a^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Ai}}^{{\prime}}}$ and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/9.9.E1
Encodings:: TeX, pMathML, png

9.9.2 $\theta\!\left(b_{{k}}\right)=\phi\!\left(b^{{\prime}}_{{k}}\right)=(k-\tfrac{1}{2})\pi.$

Symbols:: $\phi\!\left(z\right)$ : Airy phase function, $\theta\!\left(z\right)$ : Airy phase function, $b_{{k}}$ : $k$ th zero of Airy $\mathrm{Bi}$ , $b^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Bi}}^{{\prime}}}$ and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/9.9.E2
Encodings:: TeX, pMathML, png

9.9.3

${{\mathrm{Ai}}^{{\prime}}}\!\left(a_{{k}}\right)=\frac{(-1)^{{k-1}}}{\pi M\!\left(a_{{k}}\right)},$

${{\mathrm{Bi}}^{{\prime}}}\!\left(b_{{k}}\right)=\frac{(-1)^{{k-1}}}{\pi M\!\left(b_{{k}}\right)},$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\mathrm{Bi}\!\left(z\right)$ : Airy function, $M\!\left(z\right)$ : Airy modulus function, $a_{{k}}$ : $k$ th zero of Airy $\mathrm{Ai}$ , $b_{{k}}$ : $k$ th zero of Airy $\mathrm{Bi}$ and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/9.9.E3
Encodings:: TeX, TeX, pMathML, pMathML, png, png

9.9.4

$\mathrm{Ai}\!\left(a^{{\prime}}_{{k}}\right)=\frac{(-1)^{{k-1}}}{\pi N\!\left(a^{{\prime}}_{{k}}\right)},$

$\mathrm{Bi}\!\left(b^{{\prime}}_{{k}}\right)=\frac{(-1)^{{k}}}{\pi N\!\left(b^{{\prime}}_{{k}}\right)}.$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\mathrm{Bi}\!\left(z\right)$ : Airy function, $N\!\left(z\right)$ : Airy modulus function, $a^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Ai}}^{{\prime}}}$ , $b^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Bi}}^{{\prime}}}$ and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/9.9.E4
Encodings:: TeX, TeX, pMathML, pMathML, png, png

§ 9.9(iii). Derivatives With Respect to $k$

Referenced by:: Tab.9.1.1
Permalink:: http://dlmf.nist.gov/9.9.SS3

If $k$ is regarded as a continuous variable, then

9.9.5

${{\mathrm{Ai}}^{{\prime}}}\!\left(a_{{k}}\right)=(-1)^{{k-1}}\left(-\frac{da_{{k}}}{dk}\right)^{{-1/2}},$

$\mathrm{Ai}\!\left(a^{{\prime}}_{{k}}\right)=(-1)^{{k-1}}\left(a^{{\prime}}_{{k}}\frac{da^{{\prime}}_{{k}}}{dk}\right)^{{-1/2}}.$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $a_{{k}}$ : $k$ th zero of Airy $\mathrm{Ai}$ , $a^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Ai}}^{{\prime}}}$ and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/9.9.E5
Encodings:: TeX, TeX, pMathML, pMathML, png, png

See Olver (1954, Appendix).

§ 9.9(iv). Asymptotic Expansions

Notes:: See Olver (1954, Appendix).
Referenced by:: §9.17(v)
Permalink:: http://dlmf.nist.gov/9.9.SS4

For large $k$

9.9.6 $a_{{k}}=-T\left(\tfrac{3}{8}\pi(4k-1)\right),$

Symbols:: $a_{{k}}$ : $k$ th zero of Airy $\mathrm{Ai}$ , $k$ : nonnegative integer and $T$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E6
Encodings:: TeX, pMathML, png

9.9.7 ${{\mathrm{Ai}}^{{\prime}}}\!\left(a_{{k}}\right)=(-1)^{{k-1}}V\left(\tfrac{3}{8}\pi(4k-1)\right),$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $a_{{k}}$ : $k$ th zero of Airy $\mathrm{Ai}$ , $k$ : nonnegative integer and $V$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E7
Encodings:: TeX, pMathML, png

9.9.8 $a^{{\prime}}_{{k}}=-U\left(\tfrac{3}{8}\pi(4k-3)\right),$

Symbols:: $a^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Ai}}^{{\prime}}}$ , $k$ : nonnegative integer and $U$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E8
Encodings:: TeX, pMathML, png

9.9.9 $\mathrm{Ai}\!\left(a^{{\prime}}_{{k}}\right)=(-1)^{{k-1}}W\left(\tfrac{3}{8}\pi(4k-3)\right).$

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $a^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Ai}}^{{\prime}}}$ , $k$ : nonnegative integer and $W$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E9
Encodings:: TeX, pMathML, png

9.9.10 $b_{{k}}=-T\left(\tfrac{3}{8}\pi(4k-3)\right),$

Symbols:: $b_{{k}}$ : $k$ th zero of Airy $\mathrm{Bi}$ , $k$ : nonnegative integer and $T$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E10
Encodings:: TeX, pMathML, png

9.9.11 ${{\mathrm{Bi}}^{{\prime}}}\!\left(b_{{k}}\right)=(-1)^{{k-1}}V\left(\tfrac{3}{8}\pi(4k-3)\right),$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $b_{{k}}$ : $k$ th zero of Airy $\mathrm{Bi}$ , $k$ : nonnegative integer and $V$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E11
Encodings:: TeX, pMathML, png

9.9.12 $b^{{\prime}}_{{k}}=-U\left(\tfrac{3}{8}\pi(4k-1)\right),$

Symbols:: $b^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Bi}}^{{\prime}}}$ , $k$ : nonnegative integer and $U$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E12
Encodings:: TeX, pMathML, png

9.9.13 $\mathrm{Bi}\!\left(b^{{\prime}}_{{k}}\right)=(-1)^{{k}}W\left(\tfrac{3}{8}\pi(4k-1)\right).$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $b^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Bi}}^{{\prime}}}$ , $k$ : nonnegative integer and $W$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E13
Encodings:: TeX, pMathML, png

9.9.14 $\beta _{{k}}=e^{{\pi i/3}}T\left(\tfrac{3}{8}\pi(4k-1)+\tfrac{3}{4}i\ln 2\right),$

Symbols:: $\beta _{{k}}$ : $k$ th complex zero of Airy $\mathrm{Bi}$ , $k$ : nonnegative integer and $T$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E14
Encodings:: TeX, pMathML, png

9.9.15 ${{\mathrm{Bi}}^{{\prime}}}\!\left(\beta _{{k}}\right)=(-1)^{{k}}\sqrt{2}e^{{-\pi i/6}}V\left(\tfrac{3}{8}\pi(4k-1)+\tfrac{3}{4}i\ln 2\right),$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\beta _{{k}}$ : $k$ th complex zero of Airy $\mathrm{Bi}$ , $k$ : nonnegative integer and $V$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E15
Encodings:: TeX, pMathML, png

9.9.16 $\beta^{{\prime}}_{{k}}=e^{{\pi i/3}}U\left(\tfrac{3}{8}\pi(4k-3)+\tfrac{3}{4}i\ln 2\right),$

Symbols:: $\beta^{{\prime}}_{{k}}$ : $k$ th complex zero of Airy ${{\mathrm{Bi}}^{{\prime}}}$ , $k$ : nonnegative integer and $U$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E16
Encodings:: TeX, pMathML, png

9.9.17 $\mathrm{Bi}\!\left(\beta^{{\prime}}_{{k}}\right)=(-1)^{{k-1}}\sqrt{2}e^{{\pi i/6}}W\left(\tfrac{3}{8}\pi(4k-3)+\tfrac{3}{4}i\ln 2\right).$

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\beta^{{\prime}}_{{k}}$ : $k$ th complex zero of Airy ${{\mathrm{Bi}}^{{\prime}}}$ , $k$ : nonnegative integer and $W$ : expansion
Permalink:: http://dlmf.nist.gov/9.9.E17
Encodings:: TeX, pMathML, png

Here

9.9.18 $T(t)\sim t^{{2/3}}\left(1+\frac{5}{48}t^{{-2}}-\frac{5}{36}t^{{-4}}+\frac{77125}{82944}t^{{-6}}-\frac{1080\; 56875}{69\; 67296}t^{{-8}}+\frac{16\; 23755\; 96875}{3344\; 30208}t^{{-10}}-\cdots\right),$

Defines:: $T$ : expansion
Symbols:: $\sim$ : asymptotically equal
Permalink:: http://dlmf.nist.gov/9.9.E18
Encodings:: TeX, pMathML, png

9.9.19 $U(t)\sim t^{{2/3}}\left(1-\frac{7}{48}t^{{-2}}+\frac{35}{288}t^{{-4}}-\frac{1\; 81223}{2\; 0 7360}t^{{-6}}+\frac{186\; 83371}{12\; 44160}t^{{-8}}-\frac{9\; 11458\; 84361}{1911\; 0 2976}t^{{-10}}+\cdots\right),$

Defines:: $U$ : expansion
Symbols:: $\sim$ : asymptotically equal
Permalink:: http://dlmf.nist.gov/9.9.E19
Encodings:: TeX, pMathML, png

9.9.20 $V(t)\sim\pi^{{-1/2}}t^{{1/6}}\left(1+\frac{5}{48}t^{{-2}}-\frac{1525}{4608}t^{{-4}}+\frac{23\; 97875}{6\; 63552}t^{{-6}}-\frac{7\; 48989\; 40625}{8918\; 13888}t^{{-8}}+\frac{14419\; 83037\; 34375}{4\; 28070\; 66624}t^{{-10}}-\cdots\right),$

Defines:: $V$ : expansion
Symbols:: $\sim$ : asymptotically equal
Permalink:: http://dlmf.nist.gov/9.9.E20
Encodings:: TeX, pMathML, png

9.9.21 $W(t)\sim\pi^{{-1/2}}t^{{-1/6}}\left(1-\frac{7}{96}t^{{-2}}+\frac{1673}{6144}t^{{-4}}-\frac{843\; 94709}{265\; 42080}t^{{-6}}+\frac{78\; 0 2771\; 35421}{1\; 0 1921\; 58720}t^{{-8}}-\frac{20444\; 90510\; 51945}{6\; 52298\; 15808}t^{{-10}}+\cdots\right).$

Defines:: $W$ : expansion
Symbols:: $\sim$ : asymptotically equal
Permalink:: http://dlmf.nist.gov/9.9.E21
Encodings:: TeX, pMathML, png

For higher terms see Fabijonas and Olver (1999).

For error bounds for the asymptotic expansions of $a_{{k}}$ , $b_{{k}}$ , $a^{{\prime}}_{{k}}$ , and $b^{{\prime}}_{{k}}$ see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999).

§ 9.9(v). Tables

Notes:: For the computation of Tables 9.9.1–9.9.4 see 9.17(v).
Referenced by:: §9.17(v), §9.18(iv)
Permalink:: http://dlmf.nist.gov/9.9.SS5

Tables 9.9.1 and 9.9.2 give 10D values of the first ten real zeros of $\mathrm{Ai}$ , ${{\mathrm{Ai}}^{{\prime}}}$ , $\mathrm{Bi}$ , ${{\mathrm{Bi}}^{{\prime}}}$ , together with the associated values of the derivative or the function. Tables 9.9.3 and 9.9.4 give the corresponding results for the first ten complex zeros of $\mathrm{Bi}$ and ${{\mathrm{Bi}}^{{\prime}}}$ in the upper half plane.

9.9.1. Zeros of $\mathrm{Ai}$ and ${{\mathrm{Ai}}^{{\prime}}}$ .

$k$	$a_{{k}}$	${{\mathrm{Ai}}^{{\prime}}}\!\left(a_{{k}}\right)$	$a^{{\prime}}_{{k}}$	$\mathrm{Ai}\!\left(a^{{\prime}}_{{k}}\right)$
1	−2.33810 74105	0.70121 08227	−1.01879 29716	0.53565 66560
2	−4.08794 94441	−0.80311 13697	−3.24819 75822	−0.41901 54780
3	−5.52055 98281	0.86520 40259	−4.82009 92112	0.38040 64686
4	−6.78670 80901	−0.91085 07370	−6.16330 73556	−0.35790 79437
5	−7.94413 35871	0.94733 57094	−7.37217 72550	0.34230 12444
6	−9.02265 08533	−0.97792 28086	−8.48848 67340	−0.33047 62291
7	−10.04017 43416	1.00437 01227	−9.53544 90524	0.32102 22882
8	−11.00852 43037	−1.02773 86888	−10.52766 03970	−0.31318 53910
9	−11.93601 55632	1.04872 06486	−11.47505 66335	0.30651 72939
10	−12.82877 67529	−1.06779 38592	−12.38478 83718	−0.30073 08293

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $a_{{k}}$ : $k$ th zero of Airy $\mathrm{Ai}$ , $a^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Ai}}^{{\prime}}}$ and $k$ : nonnegative integer
Referenced by:: §9.9(v), §9.9(v)
Permalink:: http://dlmf.nist.gov/9.9.T1

9.9.2. Real zeros of $\mathrm{Bi}$ and ${{\mathrm{Bi}}^{{\prime}}}$ .

$k$	$b_{{k}}$	${{\mathrm{Bi}}^{{\prime}}}\!\left(b_{{k}}\right)$	$b^{{\prime}}_{{k}}$	$\mathrm{Bi}\!\left(b^{{\prime}}_{{k}}\right)$
1	−1.17371 32227	0.60195 78880	−2.29443 96826	−0.45494 43836
2	−3.27109 33028	−0.76031 01415	−4.07315 50891	0.39652 28361
3	−4.83073 78417	0.83699 10126	−5.51239 57297	−0.36796 91615
4	−6.16985 21283	−0.88947 99014	−6.78129 44460	0.34949 91168
5	−7.37676 20794	0.92998 36386	−7.94017 86892	−0.33602 62401
6	−8.49194 88465	−0.96323 44302	−9.01958 33588	0.32550 97364
7	−9.53819 43793	0.99158 63705	−10.03769 63349	−0.31693 46537
8	−10.52991 35067	−1.01638 96592	−11.00646 26677	0.30972 59408
9	−11.47695 35513	1.03849 42860	−11.93426 16450	−0.30352 76648
10	−12.38641 71386	−1.05847 18444	−12.82725 83092	0.29810 49111

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $b_{{k}}$ : $k$ th zero of Airy $\mathrm{Bi}$ , $b^{{\prime}}_{{k}}$ : $k$ th zero of Airy ${{\mathrm{Bi}}^{{\prime}}}$ and $k$ : nonnegative integer
Referenced by:: §9.9(v)
Permalink:: http://dlmf.nist.gov/9.9.T2

9.9.3. Complex zeros of $\mathrm{Bi}$ .

	$e^{{-\pi i/3}}\beta _{{k}}$		${{\mathrm{Bi}}^{{\prime}}}\!\left(\beta _{{k}}\right)$
$k$	modulus	phase	modulus	phase
1	2.35387 33809	0.09533 49591	0.99310 68457	2.64060 02521
2	4.09328 73094	0.04178 55604	1.13612 83345	−0.51328 28720
3	5.52350 35011	0.02668 05442	1.22374 37881	2.62462 83591
4	6.78865 95301	0.01958 69751	1.28822 92493	−0.51871 63829
5	7.94555 90160	0.01547 08228	1.33979 47726	2.62185 44560
6	9.02375 63663	0.01278 34808	1.38303 39005	−0.52040 69437
7	10.04106 73680	0.01089 12610	1.42042 53456	2.62071 41895
8	11.00926 72579	0.00948 68445	1.45346 64633	−0.52122 87219
9	11.93664 76131	0.00840 31785	1.48313 45656	2.62009 35195
10	12.82932 39388	0.00754 16607	1.51010 46383	−0.52171 41947

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\beta _{{k}}$ : $k$ th complex zero of Airy $\mathrm{Bi}$ and $k$ : nonnegative integer
Referenced by:: §9.9(v)
Permalink:: http://dlmf.nist.gov/9.9.T3

9.9.4. Complex zeros of ${{\mathrm{Bi}}^{{\prime}}}$ .

	$e^{{-\pi i/3}}\beta^{{\prime}}_{{k}}$		$\mathrm{Bi}\!\left(\beta^{{\prime}}_{{k}}\right)$
$k$	modulus	phase	modulus	phase
1	1.12139 32942	0.33072 66208	0.75004 14897	0.46597 78930
2	3.25690 82266	0.05938 99367	0.59221 66315	−2.63235 40329
3	4.82400 26102	0.03278 56423	0.53787 06321	0.51549 32992
4	6.16568 66408	0.02266 24588	0.50611 02160	−2.62362 85920
5	7.37383 79870	0.01731 96481	0.48406 00643	0.51928 28169
6	8.48973 85596	0.01401 65283	0.46734 68449	−2.62149 05716
7	9.53644 07072	0.01177 19311	0.45398 23240	0.52066 02139
8	10.52847 37502	0.01014 71783	0.44290 25018	−2.62052 78353
9	11.47574 11237	0.00891 66153	0.43347 44668	0.52137 15495
10	12.38537 59341	0.00795 22843	0.42529 25837	−2.61998 05803

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\beta^{{\prime}}_{{k}}$ : $k$ th complex zero of Airy ${{\mathrm{Bi}}^{{\prime}}}$ and $k$ : nonnegative integer
Referenced by:: §9.9(v), §9.9(v)
Permalink:: http://dlmf.nist.gov/9.9.T4